Let X1, X2, …, Xn be iid N(0, θ), 0<θ<∞. Show that ∑1^n Xi^2 is a sufficient statistic for θ.

Name: Problem 1, Chapter 7: Introduction to Mathematical Statistics
Uploaded: 2021-11-28T15:06:54-08:00
Duration: 6 min 37 s
Channel: Abhirup Pal
Description: Let X1, X2, …, Xn be iid N(0, θ), 0<θ<∞. Show that ∑1^n Xi^2 is a sufficient statistic for θ.

step 1 Hi, here it is given that x1, x2, xN are the n random sample which follows normal distribution w

Let X1, X2, …, Xn be iid N(0, θ), 0<θ<∞. Show that ∑1^n Xi^2...

Key Concepts

iid Random Sampling

Independent and identically distributed (iid) sampling is a fundamental assumption in many statistical models which ensures consistency in the behavior of individual data points. This assumption simplifies the structure of the likelihood function, making the identification of sufficient statistics more straightforward.

Fisher-Neyman Factorization Theorem

The Fisher-Neyman factorization theorem is a key result in statistical inference that provides a method to check sufficiency. It states that a statistic is sufficient if the likelihood function can be factored into a product where one factor depends on the parameter only through the statistic, and the other is free of the parameter.

Sufficient Statistic

A sufficient statistic is a function of the data that encompasses all the information the sample provides about a parameter. Once the sufficient statistic is known, no further details from the data can improve inference about the parameter. In this context, showing that the sum of squares captures all needed information about the variance parameter is central.

Likelihood Function

The likelihood function represents the probability of the observed data as a function of the parameter. In the case of independent and identically distributed observations from a normal distribution, constructing the likelihood function and expressing it in a factorized form is instrumental in identifying sufficient statistics.

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